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Dirac Operator

TIFRMumbai, India

Paul BaumPenn State

3 August, 2015

ATIYAH-SINGER INDEX THEOREM

There are several proofs of the Atiyah-Singer index theorem :

The first proof by M. F. Atiyah and I. M. Singer.Published in the R. Palais book Seminar on the Atiyah-SingerIndex Theorem, Princeton University Press (1965).The proof uses cobordism theory i.e. Pontrjagin-Thomconstruction and calculation of cobordism groups.A number of changes-simplifications in the proof have beenmade by M. S. Raghunathan.M. S. RaghunathanThe Atiyah-Singer Index Theorem,published in Contemporary Mathematics, Vol 522, (VectorBundles and Complex Geometry) (2008).

The K-theory proof by Atiyah-Segal-Singer.Five papers in Ann. of Math. (1968) (1971).Proves Atiyah-Singer for the equivariant caseand the families case.

The heat equation proof by Atiyah-Bott-Patodi.On the heat equation and the index theorem,Invent. Math. 19 (4): 279 - 330, (1973).Based on an idea originally proposed by McKean and Singer.Uses Chern-Weil theory.Does not in and of itself prove the general case ofAtiyah-Singer but proves a very strong result for Diracoperators. Sets the stage for the eta-invariant and indextheory on compact manifolds with boundary.

Bott periodicity proof by Baum - van Erp.Reveals exactly how and why the Atiyah-Singer index theoremcan be proved as a corollary of Bott periodicity.Does not use Pontrjagin-Thom construction and calculationof cobordism groups.

Tangent groupoid proof by Alain Connes.Published in the book Noncommutative Geometry by AlainConnes, Academic Press, (1994).See also the cyclic cohomology theory of Alain Connes.

Bivariant K-theory proof by Joachim Cuntz.Uses Kasparov bivariant K-theory.

Five lectures:

1. Dirac operator

2. Atiyah-Singer revisited

3. What is K-homology?

4. Beyond ellipticity

5. The Riemann-Roch theorem

DIRAC OPERATOR

The Dirac operator of Rn will be defined. This is a first orderelliptic differential operator with constant coefficients. Next, theclass of differentiable manifolds which come equipped with anorder one differential operator which at the symbol level is locallyisomorphic to the Dirac operator of Rn will be considered. Theseare the Spinc manifolds. Spinc is slightly stronger than oriented, soSpinc can be viewed as oriented plus epsilon. Most of theoriented manifolds that occur in practice are Spinc. The Diracoperator of a closed Spinc manifold is the basic example for theHirzebruch-Riemann-Roch theorem and the Atiyah-Singer indextheorem.

What is the Dirac operator of Rn?

To answer this, shall construct matrices E1, E2, . . . , En with thefollowing properties :

Properties of E1, E2, . . . , En

Each Ej is a 2r 2r matrix of complex numbers,

where r is the largest integer n/2.Each Ej is skew adjoint, i.e. E

j = Ej

(* = conjugate transpose)

E2j = I j = 1, 2, . . . , n(I is the 2r 2r identity matrix.)EjEk + EkEj = 0 whenever j 6= k.For n odd, (n = 2r + 1) ir+1E1E2 En = I i =

1

For n even, (n = 2r) each Ej is of the form

Ej = [ 0 0 ] and irE1E2 En =

[I 00 I

]

These matrices are constructed by a simple inductive procedure.

n = 1, E1 = [i]n n+ 1 with n odd (r r + 1)

The new matrices E1, E2, . . . , En+1 are

Ej =[

0 EjEj 0

]for j = 1, . . . , n and En+1 =

[0 II 0

]where E1, E2, . . . , En are the old matrices.n n+ 1 with n even (r does not change)

The new matrices E1, E2, . . . , En+1 are

Ej = Ej for j = 1, . . . , n and En+1 =[iI 0

0 iI

]where E1, E2, . . . , En are the old matrices.

Example

n = 1: E1 = [i]

n = 2: E1 =[

0 ii 0

], E2 =

[0 11 0

]n = 3: E1 =

[0 ii 0

], E2 =

[0 11 0

], E3 =

[i 00 i

]

Example

n = 4: E1 =

[0 0 0 i0 0 i 00 i 0 0i 0 0 0

]E2 =

[0 0 0 10 0 1 00 1 0 01 0 0 0

]

E3 =

[0 0 i 00 0 0 ii 0 0 00 i 0 0

]E4 =

[0 0 1 00 0 0 11 0 0 00 1 0 0

]

D = Dirac operator of Rn{n = 2r n evenn = 2r + 1 n odd

D =

nj=1

Ej

xj

D is an unbounded symmetric operator on the Hilbert spaceL2(Rn) L2(Rn) . . . L2(Rn) (2r times)

To begin, the domain of D isCc (Rn) Cc (Rn) . . . Cc (Rn) (2r times)

D is essentially self-adjoint(i.e. D has a unique self-adjoint extension)so it is natural to view D as an unbounded self-adjoint operatoron the Hilbert spaceL2(Rn) L2(Rn) . . . L2(Rn) (2r times)

QUESTION : Let M be a C manifold of dimension n.Does M admit a differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of Rn?

To answer this question, will define Spinc vector bundle.

What is a Spinc vector bundle?

Let X be a paracompact Hausdorff topological space.On X let E be an R vector bundle which has been oriented.i.e. the structure group of E has been reduced fromGL(n,R) to GL+(n,R)

GL+(n,R) = {[aij ] GL(n,R) | det[aij ] > 0}

n= fiber dimension (E)

Assume n 3 and recall that for n 3

H2(GL+(n,R);Z) = Z/2Z

Denote by F+(E) the principal GL+(n,R) bundle on X consistingof all positively oriented frames.

A point of F+(E) is a pair(x, (v1, v2, . . . , vn)

)where x X

and (v1, v2, . . . , vn) is a positively oriented basis of Ex. Theprojection F+(E) X is(

x, (v1, v2, . . . , vn))7 x

For x X, denote by

x : F+x (E) F+(E)

the inclusion of the fiber at x into F+(E).

Note that (with n 3)

H2(F+x (E);Z) = Z/2Z

A Spinc vector bundle on X is an R vector bundle E on X(fiber dimension E 3) with

1 E has been oriented.

2 H2(F+(E);Z) has been chosen such that x X

x() H2(F+x (E);Z) is non-zero.

Remarks

1.For n = 1, 2 E is a Spinc vector bundle =E has been orientedand an element H2(X;Z) has been chosen. ( can be zero.)

2. For all values of n = fiber dimension(E), E is a Spinc vectorbundle iff the structure group of E has been changed fromGL(n,R) to Spinc(n).i.e. Such a change of structure group is equivalent to the abovedefinition of Spinc vector bundle.

By forgetting some structure a complex vector bundle or a Spinvector bundle canonically becomes a Spinc vector bundle

complex

Spin Spinc

oriented

A Spinc structure for an R vector bundle E can be thought of asan orientation for E plus a slight extra bit of structure. Spinc

structures behave very much like orientations. For example, anorientation on two out of three R vector bundles in a short exactsequence determines an orientation on the third vector bundle. Ananalogous assertion is true for Spinc structures.

Two Out Of Three Lemma

Lemma

Let0 E E E 0

be a short exact sequence of R-vector bundles on X. If two out ofthree are Spinc vector bundles, then so is the third.

Definition

Let M be a C manifold (with or without boundary). M is aSpinc manifold iff the tangent bundle TM of M is a Spinc vectorbundle on M .

The Two Out Of Three Lemma implies that the boundary M ofa Spinc manifold M with boundary is again a Spinc manifold.

Various well-known structures on a manifold M make M into aSpinc manifold.

(complex-analytic)

(symplectic) (almost complex)

(contact) (stably almost complex)

Spin Spinc

(oriented)

A Spinc manifold can be thought of as an oriented manifold with aslight extra bit of structure. Most of the oriented manifolds whichoccur in practice are Spinc manifolds.

A Spinc manifold comes equipped with a first-order ellipticdifferential operator known as its Dirac operator. This operator islocally isomorphic (at the symbol level) to the Dirac operator ofRn.

EXAMPLE. Let M be a compact complex-analytic manifold.Set p,q = C(M,p,qT CM)p,q is the C vector space of all C differential forms of type (p, q)Dolbeault complex

0 0,0 0,1 0,2 0,n 0

The Dirac operator (of the underlying Spinc manifold) is theassembled Dolbeault complex

+ :j

0, 2j j

0, 2j+1

The index of this operator is the arithmetic genus of M i.e. isthe Euler number of the Dolbeault complex.

TWO POINTS OF VIEW ON SPINc MANIFOLDS

1. Spinc is a slight strengthening of oriented. Most of the orientedmanifolds that occur in practice are Spinc.

2. Spinc is much weaker than complex-analytic. BUT theassempled Dolbeault complex survives (as the Dirac operator).AND the Todd class survives.

M Spinc = Td(M) H(M ;Q)

If M is a Spinc manifold, then Td(M) is

Td(M) := expc1(M)/2A(M) Td(M) H(M ;Q)

If M is a complex-analyic manifold, then M has Chern classesc1, c2, . . . , cn and

expc1(M)/2A(M) = PTodd(c1, c2, . . . , cn)

WARNING!!!

The Todd class of a Spinc manifold is not obtained bycomplexifying the tangent bundle TM of M and thenapplying the Todd polynomial to the Chern classes of TCM .

Td(TCM) = A(M)2 = A(M) A(M)

Correct formula for the Todd class of a Spinc manifold M is:

Td(M) := expc1(M)/2A(M) Td(M) H(M ;Q)

SPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .

DE denotes the Dirac operator of M tensored with E.

DE : C(M,S+ E) C(M,S E)

S+, (S) are the positive (negative) spinor bundles on M .

THEOREM Index(DE) = (ch(E) Td(M))[M ].

SPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .DE denotes the Dirac

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